An additive Schwarz method for the h-p version of the finite element method in three dimensions

In this paper, we study the additive Schwarz method for the h-p version of the finite element method in three dimensions. The main idea is to treat separately the h-version (linear) components and the p-version (high-order) components by a vertex-based method. It can also be viewed as a three-level method with the level being the linear finite element approximation on the coarse mesh, the linear finite element approximation on the fine mesh, and the high-order finite element approximation on the fine mesh, respectively. The resulting algorithm can be implemented in parallel on the subdomain level for the h-version components and on the element level for the p-version components. The condition number is of order [GRAPHICS] where H-i stands for the diameter of the subdomain Omega(i), h(i) is the diameter of the elements in Omega(i), and p(i) is the maximum of the polynomial degrees used in Omega(i).